Trigonometry Sine and Cosine Values of Special Angles

Special triangles are 45-45-90 and 30-60-90 and their side lengths have special proportions that we learn by heart. To find the sine and cosine values of special angles (angles that have 0°, 30°, 45°, 60°, 90°), we use the unit circle which has a radius of one so its easy to determine these values using trigonometry. We find the sine and the cosine of all the other angles using these ones from the first quadrant. The trick is to remember which ones are positive in which quadrant. The mnemonic ASTC (All Students Take Calculus) helps us remember that.

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Hi Sir thanks for the lectures they are really helpful but can you please suggest me on which lecture to watch for this problem because I cant get my head around it.If sin(theta)=0.3,cos(x)=0.7 and tan (alpha)=0.4 then find sin(3pi/2+theta)

1 answer

Last reply by: Dr. William MurrayThu Feb 19, 2015 5:27 PM

Post by patrick guerinon February 19, 2015

Why is it that when you type in the cosine or sine of a number on a calculator, you get something like .0679 or something?

3 answers

Last reply by: Dr. William MurrayMon Nov 17, 2014 8:12 PM

Post by Tami Cumminson September 20, 2013

I know this is so simple but I really do not understand how 5pi/3 equals pi + 2pi/3.

1 answer

Last reply by: Dr. William MurrayFri Jun 21, 2013 6:22 PM

Post by A Con June 17, 2013

Please help: why, if I use a calculator to find the angle value of sin/cos/tan, does sin and tan give me the angle (neg. or pos. depending on the quadrant obviously) but cos gives me are larger angle (>90degrees), what I assume is the standard position angle?

1 answer

Last reply by: Dr. William MurrayWed May 22, 2013 3:25 PM

Post by Monis Mirzaon May 17, 2013

hi, In the extra example I, how did you know that the special triangle is 45-45-90 triangle?

Thanks

1 answer

Last reply by: Dr. William MurrayThu Nov 15, 2012 6:21 PM

Post by peter chrysanthopouloson November 14, 2012

nevermind I got it

1 answer

Last reply by: Dr. William MurrayThu Nov 15, 2012 6:20 PM

Post by peter chrysanthopouloson November 14, 2012

how did you know that it was a 30/60/90 triangle in example 1?

1 answer

Last reply by: Dr. William MurrayThu Apr 18, 2013 11:34 AM

Post by Dr. William Murrayon October 17, 2012

Hi Chin,

It depends on which direction you're going. If it's radians to degrees, multiply by 180/pi. If it's degrees to radians, multiply by pi/180.

This makes sense if you follow the rules that you learn in physics and chemistry about units: 180 degrees = pi radians, so (180 degrees)/(pi radians) = 1. Then when you want to convert in either direction, you multiply by 1:

(3 pi/4 radians) x (180 degrees)/(pi radians) = 135 degrees.

90 degrees x (pi radians)/(180 degrees) = pi/2 radians.

Hope this helps. Thanks for studying trigonometry!Will Murray

1 answer

Last reply by: Dr. William MurrayThu Apr 18, 2013 11:35 AM

Post by chin changon October 15, 2012

On the example problems you converted the degrees or radians into one or the other by multiplying pi/180 or 180/pi. How do you know what to multiply for each situation? Does it make sense? For instance on the second example problem, how did you to multiply 5pi/3 radians by 180/pi?

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Sine and Cosine Values of Special Angles

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

I have got my angles in degrees I will convert them into radians x pi/180 is equal to 5pi/6 to 10 x pi/180 is 7pi/6 radians.0127

I got those two angles in radians now, that is the first one 5pi/6, that is the second one 7pi/6.0151

And identify which quadrant each one is in, one of them is in the second quadrant, one of them is in the third quadrant, quadrant 2 and quadrant 3.0159

It all comes back to recognizing those common values, ½, square root of 3/2, square root of 2/2.0184

Once you recognize those common values, you can put these triangles in any position anywhere on the unit circle.0191

You just figure out where is your root 3/2, where is your ½, where is your root 2/2 and then you figure out which one is positive and which one is negative.0198

The whole point of this is you can figure out the sin and cos of any angle anywhere on the unit circle as long as it is a multiple of 30 or 45, or in terms of radians if it is a multiple of pi/6, pi/6, pi/4, pi/3.0209

You can figure out sin and cos of all these angles just by going back to those 3 common values and by figuring out whether their sin and cos are positive or negative.0226

Now you know how to find sin and cos of special angles, this is www.educator.com, thanks for watching.0238

The whole point of this is that you only really need to memorize the values of the triangles, root 2 over 2, root 3 over 2 and 1/2.0981

Once you know those basic triangles, you can work out what the sines and cosines are in any different quadrant just by drawing in those triangles and then figuring out which ones have to be positive, and which ones are to be negative.0991